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arXiv:2603.05486

Improved Decoding of Quantum Tanner Codes Using Generalized Check Nodes

Olai Å. Mostad, Eirik Rosnes, Hsuan-Yin Lin

quant-phcs.ITSubmitted Mar 5, 2026• Announced Mar 6, 2026• 0 citations
v1 · Mar 5, 2026v2 · Oct 5, 2026
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Abstract

We study the decoding problem for quantum Tanner codes and propose to exploit the underlying local code structure by grouping check nodes into more powerful generalized check nodes for enhanced iterative belief propagation (BP) decoding by decoding the generalized checks using a maximum a posteriori (MAP) decoder as part of the check node processing of each decoding iteration. We mainly study the finite-length setting and show that the proposed enhanced generalized BP decoder for quantum Tanner codes significantly outperforms the standard quaternary BP decoder with memory effects, as well as the recently proposed Relay-BP decoder, even outperforming generalized bicycle (GB) codes with comparable parameters in some cases. For other classes of quantum low-density parity-check (qLDPC) codes, we propose a greedy algorithm to combine checks for generalized BP decoding. However, for GB codes, bivariate bicycle codes, hypergraph product codes, and lifted-product codes, there seems to be limited gain by combining simple checks into more powerful ones. To back up our findings, we also provide a theoretical cycle analysis for the considered qLDPC codes.

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We study the decoding problem for quantum Tanner codes and propose to exploit the underlying local code structure by grouping check nodes into more powerful generalized check nodes for enhanced iterative belief propagation (BP) decoding by decoding the generalized checks using a maximum a posteriori (MAP) decoder as part of the check node processing of each decoding iteration. We mainly study the finite-length setting and show that the proposed enhanced generalized BP decoder for quantum Tanner codes significantly outperforms the standard quaternary BP decoder with memory effects, as well as the recently proposed Relay-BP decoder, even outperforming generalized bicycle (GB) codes with comparable parameters in some cases. For other classes of quantum low-density parity-check (qLDPC) codes, we propose a greedy algorithm to combine checks for generalized BP decoding. However, for GB codes, bivariate bicycle codes, hypergraph product codes, and lifted-product codes, there seems to be limited gain by combining simple checks into more powerful ones. To back up our findings, we also provide a theoretical cycle analysis for the considered qLDPC codes.
We introduce a generalized low-density parity-check decoding framework for quantum Tanner codes utilizing soft-output guessing random additive noise decoding (SOGRAND). By soft-output decoding entire component codes rather than individual parity checks, we mitigate the effects of trapping sets and cycles, resulting in improved convergence. Because our decoder preserves the message passing structure of standard belief propagation (BP), it is compatible with many BP enhancements. We demonstrate this with OSD postprocessing, quaternary BP, and RelayBP, each yielding further gains. Standalone SOGRAND outperforms the standard BP+OSD baseline by over two orders of magnitude in logical error rate. In combination with the Relay principle, SOGRAND outperforms RelayBP by over one order of magnitude, providing a way forward for scalable decoding of the emerging class of Tanner-code-based quantum codes.
The recent success in constructing asymptotically good quantum low-density parity-check (QLDPC) codes makes this family of codes a promising candidate for error-correcting schemes in quantum computing. However, conventional belief propagation (BP) decoding of QLDPC codes does not yield satisfying performance due to the presence of unavoidable short cycles in their Tanner graph and the special degeneracy phenomenon. In this work, we propose to decode QLDPC codes based on a check matrix with redundant rows, generated from linear combinations of the rows in the original check matrix. This approach yields a significant improvement in decoding performance with the additional advantage of very low decoding latency. Furthermore, we propose a novel neural belief propagation decoder based on the quaternary BP decoder of QLDPC codes which leads to further decoding performance improvements.
Generalized bicycle (GB) codes have emerged as a promising class of quantum error-correcting codes with practical decoding capabilities. While numerous asymptotically good quantum codes and quantum low-density parity-check code constructions have been proposed, their finite block-length performance often remains unquantified. In this work, we demonstrate that GB codes exhibit comparable or superior error correction performance in finite-length settings, particularly when designed with higher or unrestricted row weights. Leveraging their flexible construction, GB codes can be tailored to achieve high rates while maintaining efficient decoding. We evaluate GB codes against other leading quantum code families, such as quantum Tanner codes, single-parity-check product codes, and bivariate bicycle codes, highlighting their versatility in practical finite-length applications.
Quantum low-density parity check (QLDPC) codes are promising in realization of scalable, fault tolerant quantum memory for computation. Many of the QLDPC codes constructions suffer from unavoidable short cycles in their Tanner graph which degrade the decoding performance of the belief propagation (BP) algorithm. In this paper, we propose a syndrome based generalized belief propagation (GBP) algorithm for decoding of quantum LDPC codes and analyze how the proposed algorithm escapes from short cycle trapping sets effectively compared to the BP algorithm. Simulation results show improved decoding performance of the GBP algorithm over BP for the dual containing Calderbank, Shor and Steane (CSS) codes when cycles of length 4 are considered in the region based approach.
Quantum low-density parity-check (QLDPC) codes are among the most promising candidates for future quantum error correction schemes. However, a limited number of short to moderate-length QLDPC codes have been designed and their decoding performance is sub-optimal with a quaternary belief propagation (BP) decoder due to unavoidable short cycles in their Tanner graphs. In this paper, we propose a novel joint code and decoder design for QLDPC codes. The constructed codes have a minimum distance of about the square root of the block length. In addition, it is, to the best of our knowledge, the first QLDPC code family where BP decoding is not impaired by short cycles of length 4. This is achieved by using an ensemble BP decoder mitigating the influence of assembled short cycles. We outline two code construction methods based on classical quasi-cyclic codes and finite geometry codes. Numerical results demonstrate outstanding decoding performance over depolarizing channels.
Abstract Quantum error-correction codes are immeasurable resources for quantum computing and quantum communication. However, the existing decoders are generally incapable of checking node duplication of belief propagation (BP) on quantum low-density parity check (QLDPC) codes. Based on the probability theory in the machine learning, mathematical statistics and topological structure, a GF(4) (the Galois field is abbreviated as GF) augmented model BP decoder with Tanner graph is designed. The problem of repeated check nodes can be solved by this decoder. In simulation, when the random perturbation strength p=0.0115–0.0116 and number of attempts N = 60–70, the highest decoding efficiency of the augmented model BP decoder is obtained, and the low-loss frame error rate (FER) decreases to 7.1975 ×10−5. Hence, we design a novel augmented model decoder to compare the relationship between GF(2) and GF(4) for quantum code [[450,200]] on the depolarization channel. It can be verified that the proposed decoder provides the widely application range, and the decoding performance is better in QLDPC codes.